24576/24565: Difference between revisions
→Temperaments: added srutal archagall as a note |
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==== 2.75.85.9/7 subgroup ==== | ==== 2.75.85.9/7 subgroup ==== | ||
A fairly natural way to extend archagall is by tempering S15/S17 which [[square superparticular|(because of how semiparticulars work)]] equates two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering S15/S16 instead (equating three [[16/15]]'s with [[17/14]]), however it is unclear whether [[16/15]] can even be reached so it is preferred to think of it as adding S15/S17 = [[2025/2023]]. If you do want to reach [[16/15]] look to the next extension listed here that includes prime 5. | A fairly natural way to extend archagall is by tempering S15/S17 which [[square superparticular|(because of how semiparticulars work)]] equates two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering S15/S16 instead (equating three [[16/15]]'s with [[17/14]]), however it is unclear whether [[16/15]] can even be reached so it is preferred to think of it as adding S15/S17 = [[2025/2023]]. If you do want to reach [[16/15]] look to the next extension listed here that includes prime 5. | ||
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==== 2.3.5.7.17 subgroup (prime archagall) ==== | ==== 2.3.5.7.17 subgroup (prime archagall) ==== | ||
We may observe that in a good tuning of archagall there is an accurate [[5/4]] at +13 fourths ([[85/64]]'s) minus five octaves ([[2/1]]'s). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to [[171edo]] for which [[171edo]] is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = [[1225/1224]] and (S18/S20)/S49 = [[5832/5831]] while not tempering any of {S49, S50, S18/S20} individually. Note that [[171edo]] is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it. | We may observe that in a good tuning of archagall there is an accurate [[5/4]] at +13 fourths ([[85/64]]'s) minus five octaves ([[2/1]]'s). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to [[171edo]] for which [[171edo]] is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = [[1225/1224]] and (S18/S20)/S49 = [[5832/5831]] while not tempering any of {S49, S50, S18/S20} individually. Note that [[171edo]] is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it. | ||
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Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364 | Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364 | ||
==== Srutal archagall ==== | |||
This lower-accuracy temperament is an extension of [[srutal]] that adds prime 17 and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup. It achieves this by equating 85/64 with 4/3 by tempering their difference of S16 = 256/255. Therefore it also tempers S17 = 289/288 and thus equates 17/15 with 9/8 due to tempering S16 × S17. It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup. | |||
See [[Diaschismic family #Srutal archagall]]. | |||
=== Archagallic === | === Archagallic === | ||
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Badness: 9.335 × 10<sup>-6</sup> | Badness: 9.335 × 10<sup>-6</sup> | ||
[[Category:Mavka]] | [[Category:Mavka]] | ||